Quantum Anomalous Hall Effect in Hg 1-y Mn y Te QuantumWells

PRL 101, 146802 (2008) PHYSICAL REVIEW LETTERS 

Quantum Anomalous Hall Effect in Hg 1 yMn yTe Quantum Wells 

Chao-Xing Liu, 1,2 Xiao-Liang Qi, 2 Xi Dai, 3 Zhong Fang, 3 and Shou-Cheng Zhang 2 

1 

Center for Advanced Study, Tsinghua University, Beijing, 100084, China 

2 

Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA 

3 

Institute of Physics, Chinese Academy of Sciences, Beijing, 100080, China 

(Received 20 February 2008; published 1 October 2008) 

The quantum Hall effect is usually observed when a two-dimensional electron gas is subjected to an 

external magnetic field, so that their quantum states form Landau levels. In this work we predict that a new 

phenomenon, the quantum anomalous Hall effect, can be realized in Hg1 yMnyTe quantum wells, without 

an external magnetic field and the associated Landau levels. This effect arises purely from the spin 

polarization of the Mn atoms, and the quantized Hall conductance is predicted for a range of quantum well 

thickness and the concentration of the Mn atoms. This effect enables dissipationless charge current in 

spintronics devices. 

DOI: 10.1103/PhysRevLett.101.146802 PACS numbers: 73.43. f, 72.25.Dc, 75.50.Pp, 85.75. d 

When a two-dimensional electron gas is subjected to a 

high magnetic field, electronic states form Landau levels. 

When the temperature is low compared to the spacing 

between Landau levels, quantized Hall conductance can 

be observed. In the quantum Hall (QH) regime, electric 

current flows unidirectionally along the edge of the sample 

without any dissipation due to the absence of backscattering. 

The breaking of time reversal symmetry (TRS) is a 

necessary condition for the Hall effect; however, an external 

magnetic field (MF) is not required. Soon after the 

observation of the Hall effect, Hall also observed the 

anomalous Hall effect [1], where an additional Hall resistance 

arises from the spin-orbit interaction between electric 

current and magnetic moments. In the extreme case, an 

anomalous Hall effect can occur without the external MF 

as long as the system breaks TRS spontaneously. Given the 

experimental observation of the QH effect, it is natural to 

ask whether the anomalous Hall effect can also be quantized 

without external MF and the associated Landau levels. 

Such a question is not only of great academic interest, 

but also has important practical implications. Realizing 

dissipationless charge current through the quantum anomalous 

Hall (QAH) effect without an external MF could 

enable a new generation of quantum electronic devices. 

Some years ago, Haldane [2] constructed a theoretical 

toy model in the honeycomb lattice to show that QH effect 

is in principle possible without Landau levels. This model 

breaks TRS but conserves lattice translation symmetry. 

Haldane’s model was extended to include localization 

physics in Ref. [3]. Unfortunately, this model is mostly 

academic, and cannot be realized in the recently discovered 

graphene system. Later, the work on the intrinsic anomalous 

Hall effect of a ferromagnetic semiconductor in a 

metallic regime [4–6] showed the relation between 

Berry’s phase and Hall conductance. Qi et al. [7] constructed 

a tight-binding model of electron spin-orbit 

coupled to the polarized magnetic moments, and showed 

that Hall conductance can be quantized in appropriate 

week ending 

3 OCTOBER 2008 

parameter regimes. This model breaks TRS due to magnetic 

moments rather than Landau levels, and provides 

another example of the QAH effect. More recently, a 

closely related topological phenomenon known as the 

quantum spin Hall (QSH) effect has been theoretically 

predicted and experimentally observed in HgTe quantum 

wells (QW) [8,9]. In this work, we show that when the 

HgTe QW are doped with magnetic Mn atoms, the QAH 

effect can be realized within an experimentally accessible 

parameter regime. We propose an experiment to demonstrate 

that the quantized Hall conductance indeed arises 

from magnetic moments rather than Landau levels. 

As a starting point, we first briefly review the physics of 

the QSH effect in HgTe QW. HgTe has an inverted band 

structure, where the p-type 8 band has higher energy 

compared to the s-type 6 band at the point. For 

HgTe=CdTe QW, there exists a topological quantum phase 

transition across some critical well thickness dc where the 

band structure changes from the normal to the inverted 

character. The novel QSH effect occurs in the inverted 

regime d>dc [8]. In order to describe the physics near 

dc, an effective four-band model is introduced as 

H0ðkÞ ¼ hþðkÞ 0 

; (1) 

0 h ðkÞ 

where hþðkÞ ¼ k þ MðkÞ z þ Aðkx x ky yÞ and 

x;y;z is the Pauli matrix. h ðkÞ ¼hþð kÞ is required 

by TRS. k and MðkÞ are expanded as k ¼ C0 þ C2k2 and MðkÞ ¼M0 þ M2k2 . This effective model is expressed 

in the subspace containing the states jE1; i and 

1 

jH1; i, where jE1; i is a superposition of j 6; 2i and 

1 j 8; 2i, while jH1; i is formed by j 3 

8; 2i. Here 

denotes the two spin states which are degenerate due 

to the Kramers theorem. The diagonal block h ðkÞ describes 

a Dirac model in 2 þ 1 dimensions, which at half 

filling carries a Hall conductance of e2 =h, respectively 

[2,7,8]. Thus the net Hall conductance of the inverted QW 

system vanishes, while the spin Hall conductance, defined 

0031-9007=08=101(14)=146802(4) 146802-1 Ó 2008 The American Physical Society


as the difference between the two blocks, is still nonzero. 

Therefore the QSH effect can be viewed as two copies of 

the QAH effects, with the opposite quanta of Hall 

conductances. 

When the TRS is broken, two spin blocks are no longer 

related, and their charge Hall conductances no longer 

cancel exactly. The key idea of this work is to identify 

the parameter space where one spin block is in the normal 

regime, while the other spin block is in the inverted regime. 

The normal regime gives a topologically trivial insulator 

with vanishing Hall conductance, while the inverted regime 

gives a topologically nontrivial insulator with one 

quantum unit of Hall conductance; therefore, the whole 

system becomes a QAH state. Now we return to the fourband 

effective Hamiltonian (1) and address what kind of 

term can induce QAH effect. To describe the spin splitting 

induced by magnetization, a phenomenological term is 

introduced as 

0 

1 

GE 0 0 0 

B 0 GH 0 0 

Hs ¼ 

B 

C 

B 

C 

@ 

C 

0 0 GE 0 A 

; (2) 

0 0 0 GH where spin splitting is 2G E for the jE1; i band and 2G H 

for the jH1; i band. Then the energy gap is given by 

Eþ g ¼ 2M0 þ GE GH for the up spin block while Eg ¼ 

2M0 GE þ GH for down spin block. In order to obtain 

QAH effect, we require that (i) the state with one kind of 

spin is in the inverted regime while the other goes into the 

normal regime, namely Eþ g Eg < 0, (ii) the entire system is 

still in the insulating phase with a full bulk gap, which 

requires that jE1; þi (jE1; i) and jH1; i (jH1; þi) do 

not cross each other, leading to the condition ð2M0 þ 

GE þ GHÞð2M0 GE GHÞ > 0. Combining the above 

two conditions, we arrive at GEGH < 0, which requires 

that the spin splittings for jE1; i and jH1; i must have 

the opposite sign. This condition is illustrated in Fig. 1(a). 

We can also understand the physics from the edge state 

picture [Fig. 1(b)]. On the boundary of a QSH insulator 

there are counterpropagating edge states carrying opposite 

spin. When the spin splitting term increases, spin-down 

edge states penetrate much deeper into the bulk due to the 

decreasing gap, and eventually disappear, leaving only 

spin-up edge states bound more strongly to the edge. 

Thus the system has only spin-up edge states and transforms 

from a QSH state to a QAH state. 

Therefore the key condition GEGH < 0 is identified for 

the QAH effect. We may ask whether or not it is true in the 

realistic material. Fortunately, in HgTe QW doped with 

Mn, sp-d exchange coupling indeed gives the opposite 

signs for GE and GH, a fact which is well established in 

the literature [10,11]. From a standard perturbative treatment 

of the eight-band Kane model [8,12], we find the 

coefficients GE, GH in Eq. (2) can be expressed as GE ¼ 

ð3AF1 þ BF4Þ and GH ¼ 3B in which A, B are given 

by [11,13] A ¼ 1 

6 N0 yhSi and B ¼ 1 

6 N0 yhSi. F1, F4 are 

146802-2 

FIG. 1 (color online). Evolution of band structure and edge 

states upon increasing the spin splitting. For (a) G E < 0 and 

G H > 0, the spin-down states jE1; i and jH1; i in the same 

block of the Hamiltonian (1) touch each other and then enter the 

normal regime. But for (c) G E > 0 and G H > 0, gap closing 

occurs between jE1; þi and jH1; i bands, which belong to 

different blocks of the Hamiltonian, and thus will cross each 

other without opening a gap. In (b) we show the behavior of the 

edge states during the level crossing in the case of (a). 

1 

the amplitudes of j 6; 2i and j 8; 2i components in the 

state jE1; i, respectively, which can be extracted from the 

numerical calculation. N0 is the number of unit cells per 

unit volume, y is Mn fraction, and hSi is spin polarization 

of Mn out of the QW plane. and describe sp-d 

exchange coupling strength for the s-band and the 

p-band electron, respectively, where the signs and magnitudes 

are crucial for the relative sign of GE and GH.For Hg1 yMnyTe, these parameters are given by N0 ¼ 

0:4 eV, N0 ¼ 0:6 eV [11,13] and F1 ¼ 0:57, F4 ¼ 

0:43, leading to the opposite signs for GE and GH. Combined with the previous analysis about the 

Hamiltonian (2), we conclude that QAH effect can occur 

in MnHgTe QW, as long as Mn magnetization hSi is large 

enough and perpendicular to the QW plane. 

Although the above analysis already gives a clear explanation 

of the physical mechanism of the QAH effect, to 

obtain more quantitative predictions we perform a realistic 

electronic structure calculation based on the eight-band 

Kane model [12,13]. Our Hamiltonian takes into account 

the exchange term and bulk inversion asymmetry terms 

[12], in addition to the terms included in the original Kane 

model. In Fig. 2(a), the energy spectrum of the lowest 

subbands jE1; i and jH1; i at the point is plotted as 

a function of QW thickness d. At a critical thickness dc1 ¼ 

7:25 nm, a level crossing ‘‘A’’ occurs between jE1; i and 

1 


3 OCTOBER 2008


FIG. 2 (color online). (a) The energy levels for jE1; i and jH1; i are plotted as a function of the QW thickness. Two crossing 

points (A and B) are labeled in the figure. The energy gap ( logðE gapÞ used here) is plotted as a function of the well thickness d versus 

the Mn magnetic moment hSi in (b), versus the Mn doping concentration y in (c). Dashed blue line in (b) or (c) refers to the line along 

which (a) is plotted. The points ‘‘A’’ and ‘‘B’’ correspond to the two Dirac-type crossing points. Two different phases, conventional 

insulator (CI) with H ¼ 0 and QAH state with H ¼ e 2 =h, are separated by the gap closing line in the figures. 

jH1; i bands. This is a Dirac-type level crossing, across 

which the Hall conductance jumps by e 2 =h. Since the 

system always remains gapped when Mn magnetization is 

adiabatically turned on, we know H ¼ 0 for dd c1. The same analysis applies to the 

other critical thickness d c2 ¼ 9:4 nm[‘‘B’’ in Fig. 2(a)], 

where level crossing occurs between jE1; þi and jH1; þi 

bands and Hall conductance returns to 0. Therefore, for 

parameters hSi ¼2 and y ¼ 0:02, the QAH effect appears 

in a QW thickness range d c1 0 stands for a weak antiferromagnetic coupling between 

Mn spins. To generate the QAH effect, a small MF is 

needed to polarize Mn moments. In fact, some experiments 

have already shown quantized Hall conductance for MF 

less than 1T[9,15]. Such magnitude of MF is achievable 

through the hybrid ferromagnetic-semiconductor structure 

with a magnetic layer deposited on the top of the semicon- 

146802-3 



ductor for device application [16]. However, since any 

small MF has an orbital effect, this experiment by itself 

cannot prove the existence of QAH. To solve this problem, 

we propose two different ways to polarize Mn spins without 

an orbital MF. The first approach is to apply a lowfrequency 

polarized infrared light to provide the angular 

momentum required for aligning Mn moments. As the efficiency 

of the photoinduced Mn magnetization is quite low 

in experiment [17,18], we focus on another approach, timeresolved 

Hall measurement, which can provide a more 

dramatic demonstration of the QAH effect. First, a static 

MF is applied, which acts on itinerant electrons through 

three terms: the orbital effect H orb, the Zeeman term H Z, 

and the exchange term H ex.InH ex, Mn spin polarization is 

determined by Eq. (3). Second, at time t 0 the MF is 

switched off within a time scale B. As a result, Mn spin 

polarization will decay to zero in a spin-relaxation time s. 

However, if B s, there is a time window t 0 þ B < 

t t 0 þ s when the orbital and Zeeman effect of the MF 

already disappeared, but Mn spin polarization remains 

similar to the value before MF is removed. Consequently, 

within this time range there is no conventional QH based 

on Landau levels, so that pure QAH effect can be observed, 

if Mn spin polarization stays in the correct range. 

To illustrate this proposal more clearly, we compare 

the band structure with and without Landau levels. In 

Figs. 3(a) and 3(b), the energy spectra is plotted as a 

function of MF for different thicknesses. The blue solid 

lines take into account H orb, H Z, and H ex, corresponding to 

the spectra at time ting t 0 ¼ 0). In contrast, the 

red dashed lines are the conduction and valence band edges 

at time B


E(eV) 

(a) 

(c) 

1 

t 0 

Region A 

t 

1 

t 0 

(b) 

E(eV) 

B(T) B(T) 

Region B 

t 1=t 0 + τ s 

Region C 

of E F and MF B, three different phenomenon can be 

observed. (i) When (E F;B) is in region A of Fig. 3(a), 

the system has Hall conductance H ¼ e 2 =h in the static 

MF and enters a trivial insulator phase after MF is switched 

off. Consequently, Hall conductance will drop to zero once 

the MF is removed. (ii) When (E F;B) is in region B of 

Figs. 3(a) and 3(b), the system has Hall conductance H ¼ 

e 2 =h for static field and enters a QAH phase with the 

same Hall conductance after the MF is switched off. Thus, 

Hall conductance will remain on the e 2 =h plateau for a 

time s after turning off MF. (iii) When (E F;B)isin 

region C of Fig. 3(b), the system has vanishing Hall conductance 

in the static field, but enters a QAH phase with 

H ¼ e 2 =h after switching off MF. Consequently, we 

will observe the dramatic appearance of a ‘‘pulse’’ of Hall 

conductance in a time scale s even though the system is 

in the H ¼ 0 Hall plateau under a static field. Observation 

of this phenomenon gives the conclusive demonstration of 

the QAH effect. Because of the topological distinction 

between H ¼ 0 and H ¼ e 2 =h states, the transition between 

them is always sharp at low temperature, even 

though Mn magnetization changes continuously. Consequently, 

the time dependence of xx and xy in this proposal 

should show the same critical behavior as the usual 

‘‘plateau transition’’ in the QH effect. 

Finally, we estimate the experimental conditions required 

to realize this proposal. First, both the switchoff 

time B and Hall measurement time resolution t [19] 

should be shorter than Mn spin relaxation time s, which 

is of order 10–100 s [20,21]. Second, the MF should be 

turned off slowly enough for electrons to stay in the 

t 

1 

t 0 

t 1=t 0 + τ s 

FIG. 3 (color online). Landau level spectra are plotted as a 

function of MF (blue solid line) for two different thicknesses 

(a) d ¼ 8:5 nm and (b) d ¼ 7:6 nm, taking into account exchange 

term, Zeeman splitting, and orbital effect. The corresponding 

red dashed lines show the band edge of the subbands 

jH1; i and jE1; i when the MF has been turned off but the Mn 

spin polarization remains. Without an external MF, the system is 

in a QSH insulator phase for (a), and a trivial insulator phase 

for (b). (c) shows schematically the predicted Hall conductances 

for the three different regions A, B, and C defined by shadows 

in (a) and (b). 

t 

146802-4 



instantaneous ground state during the switchoff operation 

(adiabatic condition), which requires B @=Eg, where 

Eg is energy gap and @=Eg 10 12 s. Third, due to the 

gapless edge states, there will always be some excitation 

near the edge. Thus, we require that there is enough time 

for the electron to relax back to the equilibrium state after 

turning off MF. The time scale for relaxation is inelastic 

relaxation time, which is estimated to be 10 11 s [9]. 

Therefore, a MF of several tesla needs to be switched off 

within 10 11 

B 10 4 s, and time resolution of 

transport measurement should satisfy t 10 4 s. 

We have benefited greatly from the collaboration with 

our experimental colleagues at the University of Würzburg, 

in particular, H. Buhmann, M. König, and L. Molenkamp; 

without their generous sharing of experimental data and 

their insights on experimental feasibility this work would 

not have been possible. The authors would also like to 

thank K. Chang, T. Hughes, R. B. Liu, S. Q. Shen, J. Wang, 

F. Ye, and B. F. Zhu for helpful discussions. This work is 

supported by NSF (No. DMR-0342832) and U.S. DOE, 

Office of BES under Contract No. DE-AC03-76SF00515. 

X. D. and Z. F. acknowledge NSF of China and National 

Basic Research (973) Program of China 

(No. 2007CB925000). C.-X. L. acknowledges CSC, NSF 

(No. 10774086, 10574076), and Basic Research Development 

of China (No. 2006CB921500). 

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Quantum Anomalous Hall Effect in Hg 1-y Mn y Te QuantumWells

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